1. Introduction
Fractional calculus is that branch of classical analysis that generalizes derivatives and integrals of integer order to non-integer orders [
1,
2,
3]. There are several kinds of fractional derivatives such as the Riemann-Liouville, Caputo, Hilfer, Hadamard, and others. Recent results on the fractional calculus and fractional differential equations can be found in [
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14]. In [
15], by means of Monch’s fixed point theorem, Subashini et al. consider the existence of mild solutions to a class of evolution equations involving the Hilfer derivative.
Differential equations with impulses often serve as models in studying the dynamics of processes that are subject to sudden changes in their states. There are two popular types of impulses, namely instantaneous and non-instantaneous. In [
3], the authors studied some new classes of abstract impulsive differential equations with instantaneous impulses; for some very interesting results on equations with non-instantaneous impulses, we refer the reader to [
16,
17,
18].
The stability analysis of integral and differential equations is important in many applications, and for basic results and recent development on Ulam stability of integral and differential equations, we suggest the references [
11,
19,
20,
21,
22,
23,
24,
25].
Motivated by the results in the above mentioned papers, here we establish some new existence and stability results for the boundary value problem with nonlinear implicit generalized Hilfer-type fractional differential equations with non-instantaneous impulses
where
and
are the generalized Hilfer-type fractional derivative of order
and type
, and the generalized fractional integral of order
,
, respectively. Here,
,
,
,
,
,
,
,
,
,
and
represent the right and left hand limits of
at
,
is a given function, and
,
, are given continuous functions such that
.
Our paper is organized as follows. In
Section 2, some notations are introduced, and we recall some preliminary facts about generalized Hilfer fractional derivatives.
Section 3 contains two existence results for the problem (
1), one of which is proved using the Banach contraction principle and the other uses Krasnosel’skii’s fixed point theorem. We discuss the Ulam-Hyers-Rassias stability of our problem in
Section 4, and in
Section 5 we give an example to illustrate the applicability of our main results.
2. Preliminaries
In this section, we introduce notations, definitions, and preliminary facts that are used throughout this paper. Let
and set
. By
, we denote the Banach space of all continuous functions from
T into
with the norm
We consider the weighted Banach space
where
,
, as well as
with
We also need the Banach spaces
and
where
with the norm
We let
denote the space of complex-valued Lebesgue measurable functions
f on
for which
, where the norm is defined by
In particular, if
, the space
coincides with the space
, i.e.,
. We also wish to recall the Mittag-Leffler function given by
where
is the Euler gamma function defined by
,
. We will use the convention that
.
Definition 1 ([
26])
. (Generalized fractional integral)
Let , , and . The generalized fractional integral of order ϑ is defined by As is standard, we let denote the set of nonnegative integers.
Definition 2 ([
26])
. (Generalized fractional derivative)
Let and . The generalized fractional derivative of order ϑ is defined bywhere and Theorem 1 ([
26])
. Let , , , and . Then, for , the semigroup property is holds, i.e., Lemma 1 ([
26,
27])
. Let and . Then, is a bounded functional from into for . Lemma 2 ([
27])
. Let , , , and , . If then and Lemma 3 ([
28])
. Let , . Then, for and , we have Lemma 4 ([
27])
. Let , , and , . Then, Lemma 5 ([
27])
. Let and . If and , , then Definition 3 ([
27])
. Let the order ϑ and type r satisfy and , with . The generalized Hilfer-type fractional derivative with of a function , , is defined by In this paper, we only consider the case because .
Property 1 ([
27])
. The operator can be written as Property 2 ([
27])
. The fractional derivative is an interpolator of the following fractional derivatives: Hilfer , Hilfer–Hadamard , generalized , Caputo–type , Riemann–Liouville , Hadamard , Caputo , Caputo–Hadamard , Liouville , and Weyl . Consider the parameters
,
r, and
satisfying
We define the spaces
and
where
. Since
, it follows from Lemma 1 that
Lemma 6 ([
27])
. Let , , and . If , , then Lemma 7. Let , , and let be a function such that , , for any . Then is a solution of the differential equationif and only if x satisfies the Volterra integral equationwhere Proof. Assume
satisfies the Equation (
4) where
. We wish to show that
x is a solution of Equation (
5). From the definition of the space
, and using Lemma 1 and Definition 2, we have
From the definition of the space
, we obtain
Hence, Lemma 5 implies that
In view of Lemma 6, we see that
Therefore,
where
,
, i.e.,
x satisfies (
5).
Conversely, let
satisfy Equation (
5) for
; we need to show that
x is a solution of (
4). We apply the operator
to both sides of
, where
. Then, from Lemmas 3 and 6, we obtain
Since
, from the definition of
, we have
. Thus, (
6) implies
Now
, so from Lemma 1, it follows that
From (
7) and (
8), and the definition of the space
, we obtain
Applying the operator
to both sides of (
7) and using Lemma 5, Lemma 2, and Property 1, we have
that is, (
4) holds. This completes the proof of the lemma. □
The next two results will be used to prove our existence theorems.
Theorem 2 ([
29])
. (Banach’s fixed point theorem)
Let D be a non-empty closed subset of a Banach space E. Then any contraction mapping N of D into itself has a unique fixed point. Theorem 3 ([
29])
. (Krasnosel’skii’s fixed point theorem)
Let D be a closed, convex, and nonempty subset of a Banach space E and let A and B be operators such that: - (1)
for all ;
- (2)
A is compact and continuous;
- (3)
B is a contraction mapping.
Then there exists such that .
Next, we present the preliminaries needed in
Section 4 for the study of the Ulam stability of problem (
1). Let
,
,
, and
be a continuous function. We consider the inequalities:
and
Definition 4. Problem (1) is Ulam-Hyers (U-H) stable if there exists a real number such that for each and for each solution of inequality (9), there exists a solution of (1) with Definition 5. Problem (1) is generalized Ulam-Hyers (G.U-H) stable if there exists with such that for each and for each solution of inequality (10), there exists a solution of (1) with Definition 6. Problem (1) is Ulam-Hyers-Rassias (U-H-R) stable with respect to if there exists a real number such that for each and for each solution of inequality (11), there exists a solution of (1) with Definition 7. Problem (1) is generalized Ulam-Hyers-Rassias (G.U-H-R) stable with respect to if there exists a real number such that for each solution of inequality (11), there exists a solution of (1) with Remark 1. It is clear that:
- 1.
Definition 4 ⟹ Definition 5
- 2.
Definition 6 ⟹ Definition 7
- 3.
Definition 6 for ⟹ Definition 4
Remark 2. A function is a solution of inequality (11) if and only if there exist and a sequence , , such that: - 1.
, , , and , , ;
- 2.
, ;
- 3.
, .
Lemma 8 ([
30])
. (Gronwall type lemma) Let x and y be two integrable functions and ζ a continuous function with domain . Assume that x and y are nonnegative and ζ is nonnegative and nondecreasing. IfthenIn addition, if y is nondecreasing, then 3. Existence of Solutions
In order to study the existence of solutions to the problem (
1), we begin by considering the linear fractional differential equation
where
,
, and
, together with the conditions
where
,
,
,
,
,
, and
.
The following theorem shows that the problem (
12)–(
14) has a unique solution given by
Theorem 4. Let , where and . If , , is a function such that , then satisfies the problem (12)–(14) if and only if it satisfies (15). Proof. Assume
x satisfies (
12)–(
14). If
, then
and Lemma 7 implies that we have the solution
If
, then
. If
, then Lemma 7 implies
If
, then
, and if
, then
by Lemma 7. Repeating this process, the solution
for
can be written as
Applying
to (
16), using Lemma 3, and taking
, we obtain
Substituting in we get .
On the other hand, for
,
, applying
to
and using Lemma 3 and Theorem 1 gives
Next, taking the limit as
and using Lemma 2 with
, we obtain
Setting
in
, we see that
From
and
,
which shows that the boundary condition
is satisfied.
Now, for
,
, applying the operator
to both sides of
, and using Lemmas 3 and 6, we obtain
Since
, we have
so (
21) implies that
Since
, from Lemma 1,
From (
22) and (
23), and the definition of the space
, we obtain
Applying
to (
21) and using Lemma 5, Lemma 2, and Property 1, we have
that is, (
12) holds.
Also, it is easy to see that
and this completes the proof of the theorem. □
As a consequence of Theorem 4, we have the following result.
Corollary 1. Let where and . Let , be a function such that for any x, and . If , then x satisfies the problem (1) if and only if x is a fixed point of the operator defined bywhere φ is a function satisfying the functional equationand . In addition, . The following conditions will be used in the sequel:
- (H1)
The function
is continuous on
,
, and
- (H2)
There exist constants
and
such that
for any
x,
y,
,
and
,
.
- (H3)
The functions
are continuous and there exists a constant
such that
Remark 3. By (H2), we havewhere . We are now in a position to state and prove our first existence result for problem (
1). It is based on Banach’s fixed point theorem.
Theorem 5. Assume that conditions (H1)–(H3) hold. Ifthen the problem (1) has a unique solution in . Proof. The proof will be given in two steps.
Step 1: To show that the operator
ℑ defined in
has a unique fixed point
in
, let
x,
and
. For
, we have
and for
,
, we have
where
,
,
, such that
Therefore, for each
,
,
By Lemma 3, we have
and so
For
,
, we have
Then, for each
,
In view of (
25), this implies that the operator
ℑ is a contraction. Hence, by Theorem 2,
ℑ has a unique fixed point
.
Step 2: We need to show that the fixed point
is actually in
. Since
is the unique fixed point of operator
ℑ in
, for each
, we have
where
,
is such that
For
,
, we apply
to (
26) and using Lemmas 3 and 6, we obtain
Since
, by (H1), the right hand side is in
and thus
. Also, since
,
,
. As a consequence of Steps 1 and 2 together with Theorem 4, we can conclude that the problem (
1) has a unique solution in
. □
Our second existence result is based on Krasnosel’skii’s fixed point theorem (Theorem 3 above).
Theorem 6. In addition to conditions (H1) and (H2), assume that
- (H4)
The functions are continuous and there exist constants and such that
then the problem (1) has at least one solution in . Proof. Consider the set
where
, with
and
Define the operators
and
on
by
and
where
is a function satisfying the functional equation
The fractional integral equation
can be written as the operator equation
We shall show that the assumptions of Krasnosel’skii’s fixed point theorem are satisfied by proceeding in several steps.
Step 1: for any x, . By Remark 3 and (
24), for each
,
,
which implies that
Thus, for
, by
and Lemma 3
and for
,
, we have
Hence, for each
,
For
,
, by
and Lemma 3
we have
and for each
,
, we have
Then, for each
,
From
and
, for each
, we have
which implies that
.
Step 2: is a contraction. Let
x,
and
. By (H2), we have
where
,
,
, such that
Therefore, for
, we have
so
Then, for each
, we have
so by
, the operator
is a contraction.
Step 3: is continuous and compact. Let
be a sequence such that
in
. Then for each
,
, we have,
where
,
are such that
For each
,
, we have,
Since
, we obtain
as
for each
,
. Since the
are continuous, by the Lebesgue dominated convergence theorem, we have
Thus, is continuous.
To prove that
is uniformly bounded on
, let
. From Step 1, for each
,
This prove that the operator is uniformly bounded on .
To prove the compactness of
, take
and
. Then, for
,
,
,
where
And for
,
,
,
since the
are continuous. This proves that
is equicontinuous on
T. Therefore
is relatively compact.
By a
type Arzelà-Ascoli theorem,
is compact. As a consequence of Theorem 3,
ℑ has at least one fixed point
and in the same way as in the proof of Theorem 5, we can easily show that
. By Corollary 1, we conclude that the problem (
1) has at least one solution in the space
. □
4. Ulam-Hyers-Rassias Stability
Theorem 7. In addition to conditions (H1)–(H3) and (25), assume that - (H5)
There exist a nondecreasing function and such that for each , , we have
Then problem (1) is U-H-R stable with respect to . Proof. Let
be a solution of inequality (
11), and assume that
y is the unique solution of the problem
By Corollary 1, for each
,
where
,
, is a function satisfying the functional equation
and
. Since
x is a solution of (
11), by Remark 2, we have
Clearly, the solution of
is given by
where
, satisfies
Hence, for each
,
, we have
and by Lemma 8,
And for each
,
we have
so from (
25), we have
Then, for each
,
where
Therefore, the problem (
1) is U-H-R stable with respect to
. □
Remark 4. If conditions (H1), (H2), (H3), (H5), and (25) are satisfied, then by Theorem 7 and Remark 1, it is clear that problem (1) is U-H-R stable and G.U-H-R stable. Also, if , then problem (1) is also G.U-H stable and U-H stable. Remark 5. Our results for the boundary value problem (1) apply in the following cases. Initial value problems: , .
Anti-periodic problems: , , .
Periodic problems: , , .
5. Example
Consider the impulsive periodic generalized Hilfer fractional boundary value problem
where
,
,
,
,
, and
. Here we have
,
,
,
,
and
Clearly, , so condition (H1) is satisfied.
For each
x,
,
y,
and
, we have
so condition (H2) is satisfied with
. With
for
x,
, we have
and so condition (H3) is satisfied with
.
Also, condition (
25) holds with
Therefore, by Theorem 5, the problem (
33) has a unique solution in
.
Conditions (H5) is satisfied with
and
and
, i.e., for each
, we have
and
Theorem 7 then implies that the problem (
33) is U-H-R stable.
6. Conclusions
We provided some sufficient conditions for the existence and Ulam stability of solutions to a class of boundary value problems with nonlinear implicit non-instantaneous impulsive differential equations involving a generalized Hilfer fractional derivative. Suitable fixed point theorems were used.
Author Contributions
Conceptualization, A.S., M.B., J.R.G., and J.E.L.; Methodology, A.S., M.B., J.R.G., and J.E.L.; Formal Analysis, A.S., M.B., J.R.G., and J.E.L.; Investigation, A.S., M.B., J.R.G., and J.E.L.; Writing—Original Draft Preparation, A.S., M.B., J.R.G., and J.E.L.; Writing—Review & Editing, A.S., M.B., J.R.G., and J.E.L.; Visualization, A.S., M.B., J.R.G., and J.E.L. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Data sharing not applicable.
Conflicts of Interest
The authors declare no conflict of interest.
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